On types of elements, Gelfand and strongly harmonic rings of skew PBW extensions over weak compatible rings

. We investigate and characterize several kinds of elements such as units, idempotents, von Neumann regular, π -regular and clean elements for skew PBW extensions over weak compatible rings. We also study the notions of Gelfand and Harmonic rings for these families of algebras. The results presented here extend those corresponding in the literature for commutative and noncommutative rings of polynomial type.

[1]  A. Reyes,et al.  On Weak Annihilators and Nilpotent Associated Primes of Skew PBW Extensions , 2022, 2203.09912.

[2]  Armando Reyes,et al.  On NI and NJ skew PBW extensions , 2021, Communications in Algebra.

[3]  V. V. Bavula,et al.  Description of bi-quadratic algebras on 3 generators with PBW basis , 2021, 2112.11105.

[4]  E. Hashemi,et al.  A CLASSIFICATION OF RING ELEMENTS IN SKEW PBW EXTENSIONS OVER COMPATIBLE RINGS , 2020 .

[5]  A. Reyes,et al.  Associated prime ideals over skew PBW extensions , 2020 .

[6]  Martha Lizbeth Shaid Sandoval-Miranda,et al.  On strongly harmonic and Gelfand modules , 2018, Communications in Algebra.

[7]  O. Lezama,et al.  Skew PBW Extensions , 2020, Algebra and Applications.

[8]  A. Reyes,et al.  Skew Poincaré–Birkhoff–Witt extensions over weak compatible rings , 2020 .

[9]  José Oswaldo Lezama.  Computation of point modules of finitely semi-graded rings , 2019, Communications in Algebra.

[10]  Oswaldo Lezama,et al.  Koszulity and Point Modules of Finitely Semi-Graded Rings and Algebras , 2019, Symmetry.

[11]  A. Reyes.  Armendariz modules over skew PBW extensions , 2016, Communications in Algebra.

[12]  A. Alhevaz,et al.  EXTENSIONS OF RINGS OVER 2-PRIMAL RINGS , 2019 .

[13]  A. Tarizadeh,et al.  Characterizations of Gelfand Rings Specially Clean Rings and their Dual Rings , 2018 .

[14]  A. Reyes,et al.  A notion of compatibility for Armendariz and Baer properties over skew PBW extensions , 2017 .

[15]  A. Alhevaz,et al.  Some types of ring elements in Ore extensions over noncommutative rings , 2017 .

[16]  Oswaldo Lezama,et al.  Non-commutative algebraic geometry of semi-graded rings , 2016, Int. J. Algebra Comput..

[17]  A. Reyes,et al.  σ -PBW Extensions of Skew Armendariz Rings , 2017 .

[18]  A. Alhevaz,et al.  (Σ,∆)-Compatible Skew PBW Extension Ring , 2017 .

[19]  O. Lezama,et al.  Prime ideals of skew PBW extensions , 2014, 1402.2321.

[20]  Yueming Xiang,et al.  ORE EXTENSIONS OF SKEW π-ARMENDARIZ RINGS , 2013 .

[21]  A. Alhevaz,et al.  ON SKEW ARMENDARIZ AND SKEW QUASI-ARMENDARIZ MODULES , 2012 .

[22]  I. M. Rubio,et al.  On Spectral Compactness of Von Neumann Regular Rings , 2012 .

[23]  Chen,et al.  On Weakly Semicommutative Rings , 2011 .

[24]  Ouyang Lunqun,et al.  On Weak (α, δ)-Compatible Rings , 2011 .

[25]  O. Lezama,et al.  Gröbner Bases for Ideals of σ-PBW Extensions , 2010 .

[26]  W. Rump.  The weighted spectrum of a regular ring , 2010 .

[27]  Fanggui Wang,et al.  Gelfand Factor Rings and Weak Zariski Topologies , 2007 .

[28]  W. Tong,et al.  Spectra of Maximal 1-Sided Ideals and Primitive Ideals , 2006 .

[29]  T. Kwak,et al.  Extensions of Generalized Armendariz Rings , 2006 .

[30]  W. K. Nicholson,et al.  CLEAN RINGS: A SURVEY , 2005 .

[31]  A. Moussavi,et al.  Polynomial extensions of quasi-Baer rings , 2005 .

[32]  S. Annin.  ASSOCIATED PRIMES OVER ORE EXTENSION RINGS , 2004 .

[33]  Kenneth A. Brown,et al.  Lectures on Algebraic Quantum Groups , 2002 .

[34]  G. Marks.  ON 2-PRIMAL ORE EXTENSIONS , 2001 .

[35]  A. P. Isaev,et al.  Diffusion algebras , 2001, cond-mat/0103603.

[36]  J. McConnell,et al.  Noncommutative Noetherian Rings , 2001 .

[37]  D. Jordan.  Down–Up Algebras and Ambiskew Polynomial Rings☆ , 2000 .

[38]  Ayman Badawi.  On abelian π-regular rings , 1997 .

[39]  D. Jordan,et al.  Invariants for automorphisms of certain iterated skew polynomial rings , 1996, Proceedings of the Edinburgh Mathematical Society.

[40]  J. Krempa.  Some examples of reduced rings , 1996 .

[41]  D. Jordan.  Iterated Skew Polynomial Rings and Quantum Groups , 1993 .

[42]  Shuhao Sun.  On separation lemmas , 1992 .

[43]  Shuhao Sun.  Rings in which every prime ideal is contained in a unique maximal right ideal , 1992 .

[44]  Tsit Yuen Lam,et al.  A first course in noncommutative rings , 2002 .

[45]  Volker Weispfenning,et al.  Non-Commutative Gröbner Bases in Algebras of Solvable Type , 1990, J. Symb. Comput..

[46]  Ken R. Goodearl,et al.  Uniform rank over differential operator rings and Poincaré-Birkhoff-Witt extensions , 1988 .

[47]  H. Simmons,et al.  A Sheaf Representation for Modules with Applications to Gelfand Rings , 1984 .

[48]  Francis Borceux,et al.  Algebra in a Localic Topos with Applications to Ring Theory , 1983 .

[49]  M. Carral.  K-theory of Gelfand rings☆ , 1980 .

[50]  Christopher J. Mulvey,et al.  A generalisation of Gelfand duality , 1979 .

[51]  Christopher J. Mulvey,et al.  Representations of rings and modules , 1979 .

[52]  Ken R. Goodearl,et al.  Von Neumann regular rings , 1979 .

[53]  W. K. Nicholson.  Lifting idempotents and exchange rings , 1977 .

[54]  C. Mulvey.  A generalisation of Swan's theorem , 1976 .

[55]  Gooyong Shin.  Prime ideals and sheaf representation of a pseudo symmetric ring , 1973 .

[56]  Giuseppe De Marco,et al.  Commutative rings in which every prime ideal is contained in a unique maximal ideal , 1971 .

[57]  J. von Neumann,et al.  On Regular Rings. , 1936, Proceedings of the National Academy of Sciences of the United States of America.

[58]  O. Ore.  Theory of Non-Commutative Polynomials , 1933 .